According to the prior art, in a digital communication system, sets of consecutive bits in a bit stream are mapped to symbols, sometimes considered to be complex symbols having a real part and an imaginary part, with the real part used to modulate an in-phase carrier signal and the imaginary part used to modulate a quadrature-phase carrier signal. The data rate of such a digital communication system depends on, among other factors, the number of bits represented by each symbol. The bit error rate, depends on, among other factors, the distance, in an abstract mathematical sense, between any two points in the set of symbols used to represent/encode the bit stream. The set of symbols can be indicated as a so-called signal constellation (or symbol constellation), which is a representation showing how the correspondence between each symbol and a respective sequence of bits, the representation sometimes being provided as a plot of the symbols as points in a usually two-dimensional space. An example of a signal constellation is shown in FIG. 1A for quadrature phase shift keying (QPSK) modulation, the signal constellation there showing that the bit sequence 00 is mapped to the symbol √{square root over (P+0j)}, the bit sequence 01 is mapped to the symbol 0+j√{square root over (P)}, and so on, where P is the power used by the digital communication system in transmitting one symbol.
As mentioned above, the bit error rate of a digital communication system depends on the distance properties of the signal constellation being used. In the case of QPSK, using as a measure of distance the Euclidean between points in the abstract signal constellation space, the minimum distance between any two points in the signal constellation is, as can be seen from FIG. 1A,D=√{square root over (2P)},where, as mentioned above, P is the signal power used to transmit one symbol, i.e. the power used during one symbol period.
Another representation of the QPSK signal constellation for QPSK modulation, a representation equivalent to the representation shown in FIG. 1A, is shown in FIG. 1B. The FIG. 1B representation indicates that for the bit string corresponding to k=3 (i.e. for the bit sequence 11), the QPSK symbol is −R+0j (where R=P, as indicated in FIG. 1A).
To increase the bit rate it is possible to use so-called MSK modulation with M>4. For example, 8PSK modulation transmits 3 bits per symbol. Such approaches to improving system performance by improving the bit rate also use two-dimensional signal constellations, like QPSK modulation, but because the signal constellation space is more densely packed than for systems encoding fewer bits per symbol, the smallest distance between any two points for MSK signal constellations (with M>4) is smaller, and so the bit error rate worsens.
Another approach has been to use higher-dimensional signal constellations. The prior art teaches using signal points carved from a higher dimensional lattice/trellis code. However, the distance properties for such constellations are not optimal, and an implementation of systems using such signal constellations can be complicated.
What is needed is a higher-dimensional signal constellation not based on lattice/trellis codes, and ideally one with distance properties that are superior to those of signal constellations carved from such codes.